Gibbs Phenomenon in the Truncated Discrete-Time Fourier Transform of the Sinc Sequence

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numberofapproximationtermsN

-

π

4

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π

2

-

3π

4

-π

π

4

π

2

3π

4

π

ω

|X

N

(e

jω

)|



2

1.089

2

Using a finite number of terms of the Fourier series approximating a function gives an overshoot at a discontinuity in the function. This is called the Gibbs phenomenon. This Demonstration shows the same phenomenon with the discrete-time Fourier transform (DTFT) of a sinc sequence. The oscillations around the discontinuity persist with an amplitude of roughly 9% of the original height.